Commutator of two subgroups is normal in join
- Subgroup normalizes its commutator with any subset: If is a subgroup and is a subset of , then normalizes the following subgroup:
Here, is the commutator of the two elements.
Given: Two subgroups .
To prove: .
Proof: By fact (1), normalizes , which is the same as . Also, normalizes . Thus, the normalizer of in contains both and , hence it contains , proving that is normal in .